Identifier

etd-07062017-221901

Degree

Doctor of Philosophy (PhD)

Department

Mathematics

Document Type

Dissertation

Abstract

The Multiscale Spectral Generalized Finite Element Method (MS-GFEM) was developed in recent work by Babuska and Lipton. The method uses optimal local shape functions, optimal in the sense of the Kolmogorov n-width, to approximate solutions to a second order linear elliptic partial differential equation with L-infinity coefficients. In this dissertation an implementation of MS-GFEM over a two subdomain partition of unity is outlined and several numerical experiments are presented. The method is applied to compute local fields inside high contrast particle suspensions. The method's performance is evaluated for various examples with different contrasts between reinforcement particles and matrix material. The numerical experiments are shown to agree with a new theoretical estimate that shows the convergence rate is independent of the elastic properties of particles and matrix materials. A new domain decomposition method based on MS-GFEM is presented. Numerical computations using this iterative method are discussed and the theoretical convergence rate is provided. It is shown that the convergence rate is given by the same near-exponential bound given for MS-GFEM. A systematic method for identifying the worst case load amongst all boundary loads of a fixed energy is introduced. Here the worst case load delivers the largest fraction of input energy into a prescribed subdomain of interest. This leads to an eigenvalue problem, for which the largest eigenvalue is the maximum fraction of energy which concentrates in the subdomain. The associated eigenfunctions are the worst case solutions. These eigenfunctions are related back to the MS-GFEM shape functions and numerical results are presented for several different geometries.

Date

2017

Document Availability at the Time of Submission

Release the entire work immediately for access worldwide.

Committee Chair

Lipton, Robert P

DOI

10.31390/gradschool_dissertations.4377

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